Problem: All of the 4th grade teachers and students from Almond went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$2.50$ each for students, and the group paid $$42.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$25.50$ each for teachers and $$12.00$ each for students, and the group paid $$171.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+2.5y = 42}$ ${25.5x+12y = 171}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-25.5x-7.5y = -126}$ ${25.5x+12y = 171}$ Add the top and bottom equations together. $ 4.5y = 45 $ $ y = \dfrac{45}{4.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {8.5x+2.5y = 42}$ to find $x$ ${8.5x + 2.5}{(10)}{= 42}$ $8.5x+25 = 42$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {25.5x+12y = 171}$ and get the same answer for $x$ ${25.5x + 12}{(10)}{= 171}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.